2,934 research outputs found
Connectivity in Sub-Poisson Networks
We consider a class of point processes (pp), which we call {\em sub-Poisson};
these are pp that can be directionally-convexly () dominated by some
Poisson pp. The order has already been shown useful in comparing various
point process characteristics, including Ripley's and correlation functions as
well as shot-noise fields generated by pp, indicating in particular that
smaller in the order processes exhibit more regularity (less clustering,
less voids) in the repartition of their points. Using these results, in this
paper we study the impact of the ordering of pp on the properties of two
continuum percolation models, which have been proposed in the literature to
address macroscopic connectivity properties of large wireless networks. As the
first main result of this paper, we extend the classical result on the
existence of phase transition in the percolation of the Gilbert's graph (called
also the Boolean model), generated by a homogeneous Poisson pp, to the class of
homogeneous sub-Poisson pp. We also extend a recent result of the same nature
for the SINR graph, to sub-Poisson pp. Finally, as examples we show that the
so-called perturbed lattices are sub-Poisson. More generally, perturbed
lattices provide some spectrum of models that ranges from periodic grids,
usually considered in cellular network context, to Poisson ad-hoc networks, and
to various more clustered pp including some doubly stochastic Poisson ones.Comment: 8 pages, 10 figures, to appear in Proc. of Allerton 2010. For an
extended version see http://hal.inria.fr/inria-00497707 version
Hausdorff and packing dimensions of the images of random fields
Let be a random field with values in
. For any finite Borel measure and analytic set
, the Hausdorff and packing dimensions of the image
measure and image set are determined under certain mild
conditions. These results are applicable to Gaussian random fields,
self-similar stable random fields with stationary increments, real harmonizable
fractional L\'{e}vy fields and the Rosenblatt process.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ244 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Max-stable random sup-measures with comonotonic tail dependence
Several objects in the Extremes literature are special instances of
max-stable random sup-measures. This perspective opens connections to the
theory of random sets and the theory of risk measures and makes it possible to
extend corresponding notions and results from the literature with streamlined
proofs. In particular, it clarifies the role of Choquet random sup-measures and
their stochastic dominance property. Key tools are the LePage representation of
a max-stable random sup-measure and the dual representation of its tail
dependence functional. Properties such as complete randomness, continuity,
separability, coupling, continuous choice, invariance and transformations are
also analysed.Comment: 28 pages, 1 figur
Concentration for Coulomb gases and Coulomb transport inequalities
We study the non-asymptotic behavior of Coulomb gases in dimension two and
more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a
singular two-body interaction. We obtain concentration of measure inequalities
for the empirical distribution of such gases around their equilibrium measure,
with respect to bounded Lipschitz and Wasserstein distances. This implies
macroscopic as well as mesoscopic convergence in such distances. In particular,
we improve the concentration inequalities known for the empirical spectral
distribution of Ginibre random matrices. Our approach is remarkably simple and
bypasses the use of renormalized energy. It crucially relies on new
inequalities between probability metrics, including Coulomb transport
inequalities which can be of independent interest. Our work is inspired by the
one of Ma{\"i}da and Maurel-Segala, itself inspired by large deviations
techniques. Our approach allows to recover, extend, and simplify previous
results by Rougerie and Serfaty.Comment: Improvement on an assumption, and minor modification
Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets
We propose new summary statistics to quantify the association between the
components in coverage-reweighted moment stationary multivariate random sets
and measures. They are defined in terms of the coverage-reweighted cumulant
densities and extend classic functional statistics for stationary random closed
sets. We study the relations between these statistics and evaluate them
explicitly for a range of models. Unbiased estimators are given for all
statistics and applied to simulated examples.Comment: Added examples in version
Critical Brownian sheet does not have double points
We derive a decoupling formula for the Brownian sheet which has the following
ready consequence: An -parameter Brownian sheet in has double
points if and only if . In particular, in the critical case where ,
the Brownian sheet does not have double points. This answers an old problem in
the folklore of the subject. We also discuss some of the geometric consequences
of the mentioned decoupling, and establish a partial result concerning
-multiple points in the critical case .Comment: Published in at http://dx.doi.org/10.1214/11-AOP665 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Levy processes: Capacity and Hausdorff dimension
We use the recently-developed multiparameter theory of additive Levy
processes to establish novel connections between an arbitrary Levy process
in , and a new class of energy forms and their corresponding
capacities. We then apply these connections to solve two long-standing problems
in the folklore of the theory of Levy processes. First, we compute the
Hausdorff dimension of the image of a nonrandom linear Borel set
, where is an arbitrary Levy process in
. Our work completes the various earlier efforts of Taylor [Proc.
Cambridge Phil. Soc. 49 (1953) 31-39], McKean [Duke Math. J. 22 (1955)
229-234], Blumenthal and Getoor [Illinois J. Math. 4 (1960) 370-375, J. Math.
Mech. 10 (1961) 493-516], Millar [Z. Wahrsch. verw. Gebiete 17 (1971) 53-73],
Pruitt [J. Math. Mech. 19 (1969) 371-378], Pruitt and Taylor [Z. Wahrsch. Verw.
Gebiete 12 (1969) 267-289], Hawkes [Z. Wahrsch. verw. Gebiete 19 (1971) 90-102,
J. London Math. Soc. (2) 17 (1978) 567-576, Probab. Theory Related Fields 112
(1998) 1-11], Hendricks [Ann. Math. Stat. 43 (1972) 690-694, Ann. Probab. 1
(1973) 849-853], Kahane [Publ. Math. Orsay (83-02) (1983) 74-105, Recent
Progress in Fourier Analysis (1985b) 65-121], Becker-Kern, Meerschaert and
Scheffler [Monatsh. Math. 14 (2003) 91-101] and Khoshnevisan, Xiao and Zhong
[Ann. Probab. 31 (2003a) 1097-1141], where is computed under
various conditions on , or both.Comment: Published at http://dx.doi.org/10.1214/009117904000001026 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Clustering comparison of point processes with applications to random geometric models
In this chapter we review some examples, methods, and recent results
involving comparison of clustering properties of point processes. Our approach
is founded on some basic observations allowing us to consider void
probabilities and moment measures as two complementary tools for capturing
clustering phenomena in point processes. As might be expected, smaller values
of these characteristics indicate less clustering. Also, various global and
local functionals of random geometric models driven by point processes admit
more or less explicit bounds involving void probabilities and moment measures,
thus aiding the study of impact of clustering of the underlying point process.
When stronger tools are needed, directional convex ordering of point processes
happens to be an appropriate choice, as well as the notion of (positive or
negative) association, when comparison to the Poisson point process is
considered. We explain the relations between these tools and provide examples
of point processes admitting them. Furthermore, we sketch some recent results
obtained using the aforementioned comparison tools, regarding percolation and
coverage properties of the Boolean model, the SINR model, subgraph counts in
random geometric graphs, and more generally, U-statistics of point processes.
We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips
random complexes generated by stationary point processes. A general observation
is that many of the results derived previously for the Poisson point process
generalise to some "sub-Poisson" processes, defined as those clustering less
than the Poisson process in the sense of void probabilities and moment
measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure
- …